Frobenius objects in the category of spans

TL;DR

This paper explores Frobenius objects within the category of spans, linking them to simplicial sets and groupoids, and discusses their relevance as set-theoretic models for classical topological field theories.

Contribution

It characterizes Frobenius objects in Span via simplicial sets and connects these structures to models of topological field theories.

Findings

Frobenius objects in Span correspond to certain simplicial set data

Groupoids provide key examples of Frobenius objects in Span

Span serves as a set-theoretic model for the symplectic category

Abstract

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids. Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.